Atlas › Modular forms › Weight 12 › Level 9 › 9.12.a › 9.12.a.c

Newform orbit 9.12.a.c

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 944q^{4} + 116200q^{7} + 1128960q^{10} + 1525300q^{13} - 9876352q^{16} - 20603408q^{19} - 16128000q^{22} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{70})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} - 127575000$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 9$ )

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	0	0	0	116200	0	1525300	0	−20603408	0	0	212319016	−19148900	0	3181394800	0	0	0	−6185242196	−18227640800	0	1240285900	21236972968	0	0	263745804700

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.324 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 3, 9]: [2, 5, 12] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 2 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(9, 12)[2]   # 9.12.a.c

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 9:12 1000